Modeling and Data Analysis Associated With Supergranulation

Walter Allen

Summary

GranulationMesogranulationSupergranulationModeling SupergranulationQuestions to Be AnsweredConclusion

Granulation

Granulation is the roughening or structure of the Sun’s surface.

Granules are congregations of gas that rise to the surface from below the photosphere.

Images of granules show cascades of bright and dark areas where the dark areas are 1/4th of an arc second across.

Granules seem to oscillate in brightness with a period of 5 minutes. These oscillations help us to understand the solar interior.

Mesogranulation

Moving up in scale in terms of diameter and depth we have polygonal structures called mesogranules.

Mesogranules are hypothetical constructs and not as much are known about them.

Mesogranules are perhaps groups of individual granules with a common velocity pattern.

Supergranulation

They where first observed as 2D Doppler spectroheliograms in the 1960’s

Supergranules are large scale motion of the Sun’s surface.

It is thought that supergranules are formed from conventional granules.

Similar to conventional granulation pictures of supergranules showed dark and light regions through line of sight velocities toward and away from observation points.

The fluid motion inside and on the surface of supergranules carry the magnetic field to its boundaries. This allows the supergranule to have its characteristic polygonal shape.

Granulation

Mesogranulation

Supergranulation

Width(Mm)

Depth(Mm)

.7

5-10

25-85 10

LifeTime

10-16min.

2hrs.

>1day

Population

5,000,000

2500

Velocity(Km/sec)

~1(rms,vertical)~2(rms,horizontal)

0.3-0.5(horizontal)

0.06(rms,vertical)

.161

Definitions

Corks->Corks are a time series of magnetograms. They are composed of points that sample the speed of moving features on the surface of the Sun. Corks float above the Sun’s surface much like a cork bottle top floats on top of water.

Tree’s of Fragmenting Granules (TFG)-> TFG are families of repeatedly splitting granules originating from a single granule.

BFI (Broadband Filter Image) From 0-48hrs.

*To the left this image shows corkMotions and TFG’s (Trees of Fragmenting Granules) through the BFI (BroadbandFilter Imager)

From Hinode courtesy of Th. Roudier

*White patches are TFG with lifetimesOf 24hrs. Yellow patches are TFG withLifetimes of 20hrs. Green and blue Patches last shorter than 18hrs.

*Granules fragment and combineWith other granules to make largerStructures.

BFI (Broadband Filter Image) From 0-48hrs.

*Although TFG decrease in numberwith time a significant fraction of long lived TFG cover the Sun’s surface.

*Shorter lifetime TFG’s appearScattered everywhere in the field ofView between the longest TFG.

From Hinode courtesy of Th. Roudier

*TFG’s are tools used to quantify the temporal/spatialorganization of solar granulationat larger scales.

Stokes V Sequence 0-48hrs.

*To the left is a Stokes V imageShowing the motion of corks on aTime scale that ranges from 0 to 48hours.

*Magnetic fields are swept from the inner workings of the supergranule to its boundaries.

*Horizontal flows (fluid flow) diverge from thecell center and subside at cell boundaries.

From Hinode courtesy of Th. Roudier

Stokes V Sequence 0-48hrs.

*Cork motions are advected by theHorizontal velocity field.

*Magnetic fields are squeezed betweenTFG’s (Trees of Fragmenting Granules) and fields follow their displacement while drifting to theirboundaries.

From Hinode courtesy of Th. Roudier

*This is an image of the Stokes V Fe I at 690 nm

*TFG’s composed of parts on Mesogranule scales sweep corksAnd are pushed by new TFG.

*The sweeping of corks contribute to the formation of the larger (super-granular) scale.

From Hinode courtesy of Th. Roudier

Stokes V Fe I 630nm 0-48 hrs.

HMI (Helioseismic Magnetic Imager)

At present staff at the NSO and at Stanford have been preparing for the launch of HMI aboard SDO.

HMI will support objectives that include differential rotation, subsurface flows, magnetic flux in active regions and attributes of the tacholine.

HMI will provide space based measurements with pixel resolution of 4096 by 4096 with spherical harmonic degree of up to 1000l

Advantages Pertaining to HMI

It is hoped to use data from HMI to facilitate the modeling of supergranules.

The key goal will be to come up with a model and use the data from HMI to populate the model.

Since HMI will yield unprecedented space based observations it may prove to be the best current instrument to use.

A Dopplergram

•The Dopplergram at theleft shows the supergranulationpattern

The Model

The task is to model supergranulation.We start with a PDE which takes the form

of a wave equation.Cylindrical coordinates are used and the

boundary conditions facilitate the modeling of the supergranule.

The shape of the supergranule is generalized to that of a hexagon.

)180sin(

sin

sin

sin

CR

R

X

R XXR

Wave Equation

)()11

(2

2

2

2

22

2

2

22

t

UU

rz

U

r

U

rr

Ua

c

)()()()('),(),(),,,( tTzZMrRtzPrDtzrU

where,

Separation of Variables

22

2

22

2 11

DD

rr

D

rr

D

c

and

01 2

2

2

22

2

P

t

P

az

P

Partial Solutions

)sin()cos()(

)sin()cos()(

)sin()cos()(

)()()('

21

21

taNtaHtT

zFzGzZ

nBnBM

rYCrJCrR

cc

nn

Boundary Conditions: Horizontal

)sin

)180sin((

sin

)180sin()(

rX

JR

and

XR

R

CRnmnnm

R

CRnmnmnm

Boundary conditions in the horizontal direction are:

So that:

),,,)180sin(

sin(),,,

sin

sin(),,,( tz

XUtz

XUtzRU

CR

R

X

R

Boundary Conditions: Vertical

2222 )())sin(

)]180[sin()(

l

p

Xl

p

R

CRnmpnmnmp

0),,,(),0,,( tlrUtrU

Boundary conditions in the vertical direction are:

So that:

and

)()cos()(

cossin)(

taNtaHtT

zGzFzZ

nmnmnmnm

nmpnmnmpnm

)()}sin(]sin))sin()cos((

cos))sin()cos({[(),,,(1 00

rJztanBnB

tanAnAtzrU

nmnnmpnmpnmpnmp

nmpnmpm

nmppn

General Solution

Coefficients

drdzdzrfnrJzrJlR

A

drdzdzrfnrJzrJlR

A

drdzdzrfrJzrJlR

A

nmn

l R

nmpnmpn

nmp

nmn

l R

nmpnmpn

nmp

mn

l R

mpmp

mp

)],,()sin()()sin([)(

4

)],,()cos()()sin([)(

4

)],,()()sin([)(

2

0 021

2

0 021

2

00 0 00

21

20

Coefficients

drdzdzrgnrJzrJlR

B

drdzdzrgnrJzrJlR

B

drdzdzrgrJzrJlR

B

nmn

l R

nmpnmpn

nmp

nmn

l R

nmpnmpn

nmp

mn

l R

mpmp

mp

)],,()sin()()sin([)(

4

)],,()cos()()sin([)(

4

)],,()()sin([)(

2

0 021

2

0 021

2

00 0 00

21

20

0 2 4 6 8 10 12-0.1

0

0.1

log(r)-Km

(u)-

Km

/sec

t=1255sec, z=10000km, r=1:26000km, theta=-.33*pi:576.923:.33*pi

0 1 2 3 4 5 6 7 8-2

0

2x 10

-4

log(t)-sec

(u)-

Km

/sec

t=1:1255sec, z=10000km, r=26000km, theta=-.33*pi:576.923:.33*pi

0 1 2 3 4 5 6 7 8 9-2

0

2x 10

-4

log(z)-Km

(u)-

Km

/sec

t=1255sec, z=1:10000km. r=26000km, theta=-.33*pi:576.923:.33*pi

The plot below is generated with a sequence of numbers for r,t and z. The main goal in the future will be to populate the model with data from HMI.

0 2 4 6 8 10 12-10

0

10

log(r)-Km

log(u

)-K

m/s

ec t=1255, z=10000, r=1:26000

0 1 2 3 4 5 6 7 8-10

-5

0

log(t)-sec

log(u

)-K

m/s

ec t=1:1255, z=10000, r=26000

0 1 2 3 4 5 6 7 8 9-20

-10

0

log(z)-Km

log(u

)-K

m/s

ec t=1255, z=1:10000. r=26000

Surface Plot

-2-1

01

2

0

5000

10000-4

-2

0

2

4

x 10-5

theta-rad

t=1255sec, z=1:10000km, r=26000km, theta=-.33*pi:576.923:.33*pi

(z)-Km

U(r

,theta

,z,t

)

-3

-2

-1

0

1

2

3

x 10-5

Conclusions

For the future we seek to fulfill the following goals:I. To extend the boundary conditions so as to model

any general polygon.II. To obtain a better estimate of the depth of

supergranulation.III. To determine if supergranules are directly convective

or constructs of smaller structures.IV. To add to the governing wave equation additional

forces such as gravitational stratification and pressure gradients.

V. To take into account magneto-hydrodynamic effects.VI. To obtain clues to the origin of supergranulation.VII. To obtain a better idea of the subsurface magnetic

field.